EXPERIMENT-1
L-C-R CIRCUIT
AIM: To study resonance effect in LCR series circuit and to calculate the quality factor.
APPARATUS: A signal generator, inductor, capacitor, ammeter, resistors, AC milli voltmeter.
THEORY:
An LCR circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a
capacitor (C), connected in series or in parallel. The circuit forms a harmonic oscillator for
current, and resonates in a similar way as an LC circuit. Introducing the resistor increases the
decay of these oscillations, which is also known as damping.
(a) Series L-C-R:
When the resistor R, inductor L and the capacitor C are connected in series with a source of emf
E, the circuit is called as the series resonant or series turned circuit, shown in Figure 1
This is an acceptor circuit that means it allows maximum current to flow through it at a particular
frequency (resonant frequency) and at all other frequencies it allows less current. In A.C circuits
the voltage and the current are usually out of phase. Across the inductor, the current lags behind
the voltage is given by 90º, where as across the capacitor, the current leads the voltage by 90º.
But across the resistor the voltage and the current both are in phase. Under certain conditions, the
voltage and current are in phase, even though the circuit consists of L, C, and R and the circuit
behaves as a pure resistor. This phenomenon is called resonance. This occurs at a single
frequency known as resonance frequency. At this frequency the capacitance (XC=1/ωC) and the
inductive reactance (XL=ωL) are equal and opposite in direction. So they get cancelled each
other and only resistance acts.
The impedance of the circuit is given by Z = R + j (ωL – 1/ωC)
At resonance the reactive term disappears i.e. ωL – 1/ωC = 0
The impedance is minimum i.e Z = R
The current is maximum so ωL = 1/ωC
ω0 = ω = 2πfr = 1/√LC
At this frequency the current is maximum and is called resonance frequency. The circuit has
selective properties. To compare selectively or sharpness of resonance, the band of frequencies is
chosen at which the current falls to 1/√2 times (half power points) of its maximum value i.e Io/√2
or 0.707 Io. The difference
between the half power points is called band width(∆f)
Bandwidth:
The bandwidth is measured between the frequencies at which the power passed through the
circuit has fallen to half the value passed at resonance. There are two of these half-power
frequencies, one f= f2-f1 , where f2, is upper half power half power frequency , f1 is lower half
power frequency and f is the bandwidth
Q-factor: The Q-factor is a widespread measure used to characterize resonators. It is defined as
the peak energy stored in the circuit divided by the average energy dissipated in it per radian at
resonance. Low Q circuits are therefore damped and lossy and high Q circuits are underdamped.
Q is related to bandwidth; low Q circuits are wide band and high Q circuits are narrow band. Q=
fr/f
For a LCR circuit Q = (1/R)*√ (L/C).
PROCEDURE:
(a) Series L-C-R:
1. Connect the circuit as shown in the circuit diagram(Figure 1)
2. Apply input signal using signal generator. The output should be 10V only.
3. Take the output across the resistor and feed it to Ammeter input sockets.
4. Vary the frequency till the Ammeter records a sharp rise and fall, adjust the signal such that
the Ammeter deflection is the maximum possible. This is the resonant frequency of the
connected combination of the circuit.
5. Adjust the signal generator amplitude such that to get full-scale deflection. In Ammeter now
reduce the frequency till the deflection falls considerably. Then increase the frequency in regular
intervals & note down the Ammeter readings.
6. Plot a graph between the meter deflection divisions and frequency.
7. Repeat the procedure using different combinations of L, C & R and study how Q is affected.
Also study how Resonant Frequency depends upon different combination of L.C.R.
L
C
R
A
SIGNAL
GENERATOR
Figure 1. LCR series circuit
OBSERVATION TABLE 1: L-C-R Series
L = _______ mH C = _______µF.
S.No
R= ________Ω
Frequency (Hz)
Current(mA)
FORMULA:
1) Resonant frequency
fr 
1
2 LC
Where L = inductance of the coil, C = Capacitance of the
capacitor
2) Quality factor
fr
Q =_______ Where Bandwidth ∆f = f1 – f2 (Hz)
∆f
fr = resonant frequency of the series resonant circuit.
A graph is drawn for current against frequency. The frequency corresponding to maximum
current is noted and it is the resonant frequency fr. The frequencies f1and f2 corresponding to half
power points is noted and from it the band width, (f1-f2) is noted. From the values of fr, f1 and f2
the quality factor, Q is calculated.
MODEL GRAPH
RESULT
.
For a Series resonance circuit:
1)The resonant frequency fr = ___________Hz
2) The Bandwidth ∆f = _______ (Hz)
3) Quality factor Q =
__________.